Yes, and the thing is that many of the trappings of at least the Catholic church reflect a strong "Romanization" of the religion and the fusion of various "pagan" elements (stylistic and ritualistic) of Roman or Mithraic pagainism into the practice. Vestal virgins to nuns, etc.
Christianity without a powerful pan-Roman culture may have stayed as a kind of messaniac Jewish sect, confined to Jewish communities around the mediteranean.
Perhaps the appeal of Christianity was just happenstance or perhaps it hit a sweet spot that would have worked well even a large empire (with a large number of slaves etc) that would have originated from a Punic civilization and its religion, myths, culture.
It would have likely looked very different from this Christianity, possibly to the point that we'd barely recognize any similarity.
I'm no expert, but think the Coptic sects accepted early Roman orthodoxy, participated in the various councils, adopted the Nicene creed, condemned the various heresies, etc. So they differ extensively from the rather heterogenous beliefs and practices of the church of 1st and 2nd century.
In a way, maybe Islam is closer to what you're suggesting. A monotheistic religion without the dominance of the Roman empire or influence from it, and without the rather odd violent debates about the trinity and the relationship of Christ to God etc. that went on in early Church
Christianity without a powerful pan-Roman culture may have stayed as a kind of messaniac Jewish sect, confined to Jewish communities around the mediteranean.